Efficient Spin Polarization

ABSTRACT

In some aspects, polarization of a spin ensemble can be increased using cavity-based techniques. A cavity is coupled with a spin ensemble, and a drive field generates an interaction between the cavity and the spin ensemble. In some cases, the interaction increases the polarization of the spin ensemble faster than the thermal (T 1 ) relaxation process or any other thermal polarizing process affecting the spin ensemble. In some cases, polarization is increased by iteratively acting on angular momentum subspaces of the spin ensemble, and mixing the angular momentum subspaces, for example, by a dipolar interaction, a transverse (T 2 ) relaxation process, application of a gradient field, or a combination of these and other processes.

CLAIM OF PRIORITY

This application claims priority to U.S. Provisional Patent ApplicationSer. No. 61/819,103, filed on May 3, 2013, the entire contents of whichare hereby incorporated by reference.

BACKGROUND

This document relates to using a cavity to increase spin polarization inmagnetic resonance applications.

In magnetic resonance systems, signal-to-noise ratio (SNR) generallydepends on the spin polarization and the time required to reach thermalequilibrium with the environment. The time required to reach thermalequilibrium—characterized by the energy relaxation time T₁—often becomeslong, for example, at low temperatures. Conventional techniques forremoving entropy from a quantum system include dynamic nuclearpolarization (DNP), algorithmic cooling, optical pumping, laser cooling,and microwave cooling, among others.

Various approaches have been used to increase the signal-to-noise ratio(SNR) in magnetic resonance applications. For instance, signal averagingover multiple acquisitions is often used to increase SNR. Anotherapproach is to increase the induction probe sensitivity, for example, byoverlapping multiple induction coils and using phased array techniques.In some systems, induction probes are embedded in cryogens to reduceintrinsic noise within the induction probes.

SUMMARY

In some aspects, a cavity is coupled with a spin ensemble, and aresonator generates a drive field that produces an interaction betweenthe cavity and the spin ensemble. In some cases, the interactionincreases the polarization of the spin ensemble faster than the thermal(T₁) relaxation process or any other thermal polarizing processaffecting the spin ensemble.

In some implementations, the cavity is detuned from the spin-resonancefrequency of the spin ensemble, and the Rabi frequency associated withthe drive field is matched to the cavity detuning. In some cases, thespin ensemble's polarization is increased iteratively, for example, bycavity-based cooling acting independently on each angular momentumsubspace of the spin ensemble, and a mixing process mixing the angularmomentum subspaces.

In some implementations, cavity-based cooling can be made availableon-demand and provide faster than thermal polarization. Increasing thespin ensemble's polarization may lead to an improved SNR, or otheradvantages in some cases.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features,objects, and advantages will be apparent from the description anddrawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1A is a schematic diagram of an example magnetic resonance system.

FIG. 1B is a schematic diagram of an example control system.

FIG. 1C is a flow chart of an example technique for increasingpolarization of a spin ensemble.

FIG. 2 is a plot showing a spin-resonance frequency, a cavity-resonancefrequency, and a Rabi frequency in an example magnetic resonance system.

FIG. 3 shows two example energy level diagrams for a spin coupled to atwo-level cavity.

FIG. 4 is a plot showing simulated evolution of the normalizedexpectation value of −(J_(x)(t))/J for the Dicke subspace of an examplecavity-cooled spin ensemble.

FIG. 5 is an energy level diagram of an example spin system coupled to atwo-level cavity.

FIG. 6 is a diagram of an example 3-spin Hilbert space.

FIG. 7 is a plot showing effective cooling times calculated for examplespin ensembles.

FIG. 8A is a schematic diagram showing entropy flow in an examplecavity-based cooling process.

FIG. 8B is a plot showing example values of the rates Γ_(SC) and Γ_(CF)shown in FIG. 8A.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Here we describe techniques that can be used, for example, to increasethe signal-to-noise ratio (SNR) in a magnetic resonance system byrapidly polarizing a spin ensemble. The techniques we describe can beused to achieve these and other advantages in a variety of contexts,including nuclear magnetic resonance (NMR) spectroscopy, electron spinresonance (ESR) spectroscopy, nuclear quadrupole resonance (NQR)spectroscopy, magnetic resonance imaging (MRI), quantum technologies anddevices, and other applications.

We describe cavity-based cooling techniques applied to ensemble spinsystems in a magnetic resonance environment. In some implementations, acavity having a low mode volume and a high quality factor is used toactively drive all coupled angular momentum subspaces of an ensemblespin system to a state with purity equal to that of the cavity on atimescale related to the cavity parameter. In some instances, byalternating cavity-based cooling with a mixing of the angular momentumsubspaces, the spin ensemble will approach the purity of the cavity in atimescale that can be significantly shorter than the characteristicthermal relaxation time of the spins (T₁). In some cases, the increasein the spin ensemble's polarization over time during the cavity-basedcooling process can be modeled analogously to the thermal spin-latticerelaxation process, with an effective polarization rate (1/T_(1,eff))that is faster than the thermal relaxation rate (1/T₁).

Accordingly, the cavity can be used to remove heat from the spinensemble (reducing the spin temperature) or to add heat to the spinensemble (increasing the spin temperature), thereby increasing the spinpolarization. Heating the spin ensemble can create an invertedpolarization, which may correspond to a negative spin temperature.

FIG. 1A is a schematic diagram of an example magnetic resonance system100. The example magnetic resonance system 100 shown in FIG. 1A includesa primary magnet system 102, a cooling system 120, a resonator andcavity system 112, a sample 110 that contains spins 108, and a controlsystem 118. A magnetic resonance system may include additional ordifferent features, and the components of a magnetic resonance systemcan be arranged as shown in FIG. 1A or in another manner.

The example resonator and cavity system 112 can be used to control thespin ensemble as described in more detail below. In some cases, thecavity and resonator system 112 increases polarization of the spinensemble by heating or cooling the spin ensemble.

The cooling system 120 provides a thermal environment for the resonatorand cavity system 112. In some cases, the cooling system 120 can absorbheat from the cavity to maintain a low temperature of the cavity. Thecooling system 120 may reside in thermal contact with the resonator andcavity system 112, the sample 110, or both. In some cases, the coolingsystem 120 maintains the resonator and cavity system 112, the sample110, or both at liquid helium temperatures (e.g., approximately 4Kelvin), at liquid nitrogen temperatures (e.g., approximately 77Kelvin), or at another cryogenic temperature (e.g., less than 100Kelvin). In some cases, the cooling system 120 maintains the resonatorand cavity system 112, the sample 110, or both at pulsed-tuberefrigerator temperatures (e.g., 5-11 Kelvin), pumped helium cryostattemperatures (e.g., 1.5 Kelvin), helium-3 fridge temperatures (e.g., 300milliKelvin), dilution refrigerator temperatures (e.g., 15 milliKelvin),or another temperature.

In some cases, the resonator and the cavity are implemented as twoseparate structures, and both are held at the same cryogenictemperature. In some cases, the resonator and the cavity are implementedas two separate structures, and the cavity is held at a cryogenictemperature while the resonator is held at a higher temperature. In somecases, an integrated resonator/cavity system is held at a cryogenictemperature. In general, various cooling systems can be used, and thefeatures of the cooling system 120 can be adapted for a desiredoperating temperature T_(C), for parameters of the resonator and cavitysystem 112, or for other aspects of the magnetic resonance system 100.

In some implementations, the resonator and cavity system 112 operates ata desired operating temperature T_(C) that is in the range from roomtemperature (approximately 300 K) to liquid helium temperature(approximately 4 K), and the cooling system 120 uses liquid-flowcryostats to maintain the desired operating temperature T_(C). Thecooling system 120 can include an evacuated cryostat, and the resonatorand cavity system 112 can be mounted on a cold plate inside thecryostat. The resonator and cavity system 112 can be mounted in thermalcontact with the cryostat, and it can be surrounded by a thermalradiation shield. The cooling system 120 can be connected to a liquidcryogen source (e.g., a liquid nitrogen or liquid helium Dewar) by atransfer line, through which the liquid cryogen can be continuouslytransferred to the cold head. The flow rate and liquid cryogen used cancontrol the operating temperature. Gases can be vented through a vent.

In some cases, the cooling system 120 uses a closed-loop system (e.g., acommercial Gifford-McMahon pulsed-tube cryo-cooler) to maintain thedesired operating temperature T_(C) of the resonator and cavity system112. A closed-loop or pulsed-tube system may, in some instances, avoidthe need for continuously transferring costly liquid cryogen. In someclosed-loop refrigerators, the cryostat has two-stages: the first stage(ranging, e.g., from 40 to 80 K) acts as a thermal insulator for thesecond stage, and the second stage encases the cold head and theresonator and cavity system 112. Some example closed-loop systems canreach a stable operating temperature of 10 Kelvin.

In some cases, the cooling system 120 uses a liquid helium cryostat tomaintain the desired operating temperature T_(C) of the resonator andcavity system 112. A liquid helium cryostat can be less complicated andmore stable in some applications. When a liquid helium cryostat is usedthe resonator and cavity system 112 can be immersed (e.g., fully orpartially immersed) in liquid helium. The system can include an outerDewar that contains liquid nitrogen and an inner Dewar that containsliquid helium, and the two Dewars can be separated by a vacuum jacket oranother thermal insulator. Liquid helium cryostat systems can typicallyreach a stable operating temperature of approximately 4 Kelvin.

In some cases, the cooling system 120 uses a helium-gas-flow (orpumped-helium) cryostat to maintain the desired operating temperatureT_(C) of the resonator and cavity system 112. Some commercialhelium-gas-flow (or pumped-helium) cryostats can reach a stableoperating temperature of 1.5 Kelvin. In such cases, the resonator andcavity system 112 can be mounted inside the cryostat, and a flow ofhelium gas can be communicated over the surface of the resonator andcavity system 112. In some implementations, the cooling system 120includes a liquid helium Dewar that surrounds the resonator and cavitysystem 112 and is thermally isolated by a vacuum jacket, and a valve(e.g., a mechanically-controlled needle valve in the liquid heliumDewar) can control the flow of helium from the Dewar. The valve cancontrol a port that opens into a gas heater, so that the liquid heliumis vaporized and flows to the resonator and cavity system 112. The valveand heater can be externally controlled to provide the desiredtemperature regulation.

Some example helium-gas-flow cryostats can reach operating temperaturesof 1 Kelvin by lowering the vapor pressure of the helium gas in thecryostat. This can be achieved by pumping on the helium in a smallcontainer (known as the “1-K pot”) inside the vessel to lower the vaporpressure and thereby lower the boiling point of liquid helium (e.g.,from 4.2 Kelvin down to 1 Kelvin). Some systems can cool down evenfurther and reach milliKelvin temperatures, for example, using thehelium-3 isotope (which is generally more expensive than the helium-4isotope). The helium-3 can be pumped to much lower vapor pressures,thereby lowering the boiling point as low as 200 milliKelvin. Aclosed-loop system can be used to avoid leaks and preserve the helium-3material.

In some cases, the cooling system 120 uses a dilution refrigeratorsystem to maintain the desired operating temperature T_(C) of theresonator and cavity system 112. Dilution refrigerator systems typicallyuse a helium-3 circulation system that is similar to the helium-gas-flowcryostat described above. The dilution fridge system can pre-cool thehelium-3 before entering the 1-K pot, to provide an operatingtemperature as low as 2 milliKelvin.

The magnetic resonance system 100 shown in FIG. 1A can polarize the spinensemble in the sample 110. For example, the magnetic resonance system100 can cool or map the spin ensemble to a thermal equilibrium state orto another state (i.e., a state other than the thermal equilibriumstate, which may be more polarized or less polarized than the thermalequilibrium state).

In the example shown, the spins 108 in the sample 110 interactindependently with the primary magnet system 102 and the resonator andcavity system 112. The primary magnet system 102 quantizes the spinstates and sets the Larmor frequency of the spin ensemble. Rotation ofthe spin magnetization can be achieved, for example, by aradio-frequency magnetic field generated by a resonator. While the spinsare weakly coupled to the environment, the cavity is well coupled to theenvironment (e.g., the cooling system 120) so that the time it takes forthe cavity to reach thermal equilibrium is much shorter than the time ittakes the spins to reach thermal equilibrium. The resonator can driveRabi oscillations in the spin ensemble so that they couple to thecavity, and the Dicke states and other angular momenta subspaces of thespin system reach thermal equilibrium with the cavity.

The resonator and cavity system 112 can be described in terms of acavity resonance and a spin resonance. The spin resonance is shiftedfrom the cavity resonance by the Rabi frequency. The Rabi frequency(i.e., the frequency of the Rabi oscillations) can be a function of thepower of the drive field applied at the spin-resonance frequency. TheRabi frequency can be configured to couple the spins to the cavitymodes. For example, the power of the drive field can be set such thatthe Rabi frequency is substantially equal to the difference between thecavity resonance and the spin resonance. In some cases, the system canbe modeled as a set of Dicke states and angular momenta subspaces of thespin ensemble (i.e., states in the Dicke and angular momenta subspace)coupled to the cavity modes through the Tavis-Cummings Hamiltonian.

A cavity having a low mode volume and high quality factor can produce astrong spin-cavity coupling for the spin ensemble. In some instances,the rate of photon exchange between the Dicke states and cavity scalesas √{square root over (N_(s))} (the number of spins in the spinensemble) and g (the spin-cavity coupling strength for a single spin).In some examples, the spin-cavity coupling strength is inverselyproportional to the square root of the mode volume and directlyproportional to the square root of the admittance (i.e., the qualityfactor of the cavity).

In some implementations, the cavity is cooled efficiently and quickly,and the heat capacity of the cavity is large compared to the heatcapacity of the spins. In some instances, the polarization rate producedby the spin-cavity interaction can be significantly faster than thethermal T₁ relaxation process. In some cases, the polarization rateproduced by the spin-cavity interaction is faster than any internalrelaxation process affecting the spin ensemble, including spontaneousemission, stimulated emission, thermal T₁ relaxation, or others. Forexample, as a result of the low mode volume and high quality factorcavity, the efficient cavity cooling, the efficient spin-cavitycoupling, the mixing of angular momenta subspaces or a combination ofthese and other features, the spin ensemble can be cooled quickly towardthe ground state. The mixing of angular momenta subspaces can beachieved, for example, by repeating a cavity-cooling process and usingan interaction such as the Dipolar coupling, natural T₂ relaxation,external gradient fields, etc. In some aspects, this can provide aneffective “short circuit” of the T₁ relaxation process. For example, thetechnique shown in FIG. 1C can be used to achieve faster spinpolarization in some instances.

FIG. 1C is a flow chart showing an example process 195 for increasingpolarization of a spin ensemble. The example process 195 can beperformed, for example, in the example magnetic resonance system 100shown in FIG. 1A or in another type of system. The example process 195shown in FIG. 1C can include additional or different operations. In somecases, individual operations can be divided into multiplesub-operations, or two or more of the operations can be combined orperformed concurrently as a single operation. Moreover, some or all ofthe operations can be iterated or repeated, for example, until a desiredstate or polarization is achieved or until a terminating condition isreached.

As shown in FIG. 1C, at 196, angular momenta subspaces of a spinensemble are mapped to a lower-energy state. For example, one or moreangular momenta subspaces may be cooled to their respective loweststates. In some cases, a coherent interaction between the cavity and thespin ensemble can drive each angular momentum subspace to its lowestenergy state. The mapping can be generated, for example, by applying adrive field to the spin ensemble. At 197, the angular momenta subspacesare connected. One or more of a number of different techniques can beused to connect the angular momenta subspaces. In some instances, theangular momenta subspaces are connected by a process that mixes thevarious subspaces of the overall space. For example, a dipolarinteraction among spins, transverse (T₂) relaxation, an externalgradient field, a similar external or internal dephasing interaction, ora combination of one or more of these can be used to connect the angularmomenta subspaces. At 198, a more highly-polarized state is obtained.That is to say, the state of the spin ensemble can be more highlypolarized than before the spin ensemble's angular momenta subspaces werecooled to their respective lowest states (at 196) and connected (at197). The operations (196, 197) can be iterated one or more times, forexample, until a desired polarization or other condition is reached.

In some implementations, the initial state of the spin ensemble (before196) has less polarization than the spin ensemble's thermal equilibriumstate. For example, the initial state of the spin ensemble may be ahighly mixed state that has little or no polarization. The polarizationof the state produced on each iteration can be higher than thepolarization of the initial state. In some instances, the polarizationis subsequently increased on each iteration. For example, the operations(196, 197) may be repeated until the spin ensemble reaches a thermalequilibrium polarization or another specified polarization level (e.g.,an input polarization for a magnetic resonance sequence to be applied tothe spin ensemble).

In some implementations, the process 195 can be used to polarize a spinensemble on-demand. For example, the process 195 can be initiated at anytime while the sample is positioned in the magnetic resonance system. Insome cases, the spin ensemble is polarized between imaging scans orbetween signal acquisitions. Generally, the spin ensemble can be in anystate (e.g., any fully or partially mixed state) when the process 195 isinitiated. In some cases, the process 195 is initiated on-demand at aspecified time, for example, in a pulse sequence, a spectroscopy orimaging process, or another process, by applying the Rabi field for aspecified amount time.

In the example shown in FIG. 1A, the spin ensemble can be any collectionof particles having non-zero spin that interact magnetically with theapplied fields of the magnetic resonance system 100. For example, thespin ensemble can include nuclear spins, electron spins, or acombination of nuclear and electron spins. Examples of nuclear spinsinclude hydrogen nuclei (¹H), carbon-13 nuclei (¹³C), and others. Insome implementations, the spin ensemble is a collection of identicalspin-½ particles.

The example primary magnet system 102 generates a static, uniformmagnetic field, labeled in FIG. 1A and referred here to as the B₀ field104. The example primary magnet system 102 shown in FIG. 1A can beimplemented as a superconducting solenoid, an electromagnet, a permanentmagnet or another type of magnet that generates a static magnetic field.In FIG. 1A, the example B₀ field 104 is homogeneous over the volume ofthe sample 110 and oriented along the z direction (also referred to hereas the “axial direction”) of the axisymmetric reference system 106.

In the example system shown in FIG. 1A, interaction between the spins108 and the primary magnet system 102 includes the Zeeman HamiltonianH=−μ·B, where μ represents the magnetic moment of the spin and Brepresents the magnetic field. For a spin-½ particle, there are twostates: the state where the spin is aligned with the B₀ field 104, andthe state where the spin is anti-aligned with the B₀ field 104. With theB₀ field 104 oriented along the z-axis, the Zeeman Hamiltonian can bewritten H=−μ_(z)B₀. Quantum mechanically, μ_(z)=γσ_(z) where γ is thespin gyromagnetic ratio and σ_(z) is the z-direction spin angularmomentum operator with angular momentum eigenstates |m

_(s) and eigenvalues m=±½, where  is Planck's constant. The factorω_(S)=γB₀ is the spin-resonance frequency also known as the Larmorfrequency.

In the example shown in FIG. 1A, the thermal distribution of individualmembers of the ensemble being either aligned or anti-aligned with the B₀field 104 is governed by Maxwell-Boltzmann statistics, and the densitymatrix for the thermal equilibrium state is given by

${\rho = {\frac{1}{z}^{{- H}/{kT}}}},$

where the denominator

is the partition function, and H is the Hamiltonian of the spinensemble. The partition function can be expressed

=Σe^(−H/kT,) where the sum is over all possible spin ensembleconfigurations. The constant k is the Boltzmann factor and T is theambient temperature. As such, the thermal equilibrium state of the spinensemble (and the associated thermal equilibrium polarization) can bedetermined at least partially by the sample environment (including themagnetic field strength and the sample temperature), according to theequation above. The polarization of the spin ensemble can be computed,for example, from the density matrix representing the state of the spinensemble. In some instances, the spin polarization in the z-directioncan be computed as the expectation value of the magnetization in thez-direction, M_(Z), as follows:

M _(Z)

=(γ)Tr{J _(Z)ρ}

where J_(Z)≡Σ_(j=1) ^(N) ^(S) σ_(Z) ^((j))/2 is the total spin ensemblez-angular momentum and N_(s) is the ensemble spin size.

Once the spin ensemble has thermalized with its environment, anyexcitations that cause deviations away from thermal equilibrium willnaturally take time (characterized by the thermal relaxation rate T₁) tothermalize. The thermal relaxation process evolves the spin ensemblefrom a non-thermal state toward the thermal equilibrium state at anexponential rate that is proportional to 1/T₁. Many magnetic resonanceapplications manipulate the spins and acquire the inductive signalsgenerated by them. Signal averaging is customarily used to improve thesignal-to-noise ratio (SNR). However, the relaxation time T₁ may berelatively long and the efficiency of signal averaging is therebyreduced. In the example shown in FIG. 1A, the resonator and cavitysystem 112 can be used (e.g., in the example process 195 shown in FIG.1C, or in another manner) to effectively “short-circuit” the relaxationprocess, which significantly reduces this wait time and increases theefficiency of signal averaging.

In some instances, the resonator and cavity system 112 can include aresonator component that controls the spin ensemble, and a cavitycomponent that cools the spin ensemble. The resonator and cavity can beimplemented as separate structures, or an integrated resonator/cavitysystem can be used. In some implementations, the resonator is tuned to aresonance frequency of one or more of the spins 108 in the sample 110.For example, the resonator can be a radio-frequency resonator, amicrowave resonator, or another type of resonator.

The resonator and cavity system 112 is an example of a multi-moderesonance system. In some examples, a multi-mode resonance system hasone or more drive frequencies, one or more cavity modes, and possiblyother resonance frequencies or modes. The drive frequency can be tunedto the spins' resonance frequency, which is determined by the strengthof the B₀ field 104 and the gyromagnetic ratio of the spins 108; thecavity mode can be shifted from the drive frequency. In some multi-moderesonance systems, the drive frequency and the cavity mode are providedby a single integrated structure. Examples of integrated multi-moderesonator structures include double-loop resonators, birdcageresonators, and other types of structures. In some multi-mode resonancesystems, the drive frequency and cavity mode are provided by distinctstructures. In some cases, the geometry of a low quality factor (low-Q)coil can be integrated with a high-Q cavity such that both the coil andcavity are coupled to the spin system but not to each other. Thetechniques described here can operate using a single drive frequency orpossibly multiple drive frequencies applied to the coil.

In the example shown in FIG. 1A, the cavity has a resonance frequencya), that is different from the resonance frequency of the resonator. Thecavity of the example resonator and cavity system 112 supportselectromagnetic waves whose modes are determined by physicalcharacteristics of the cavity. Typically, the fundamental mode is usedas the cavity resonance and the quality factor of the cavity (Q) can bedefined as the ratio of the stored energy in the cavity mode to thedissipated energy. In terms of frequency units, the quality factor ofthe cavity may be represented

${Q = \frac{\omega_{c}}{\Delta \; \omega}},$

where ω_(c) is the cavity-resonance frequency, and Δω is the −3 dBbandwidth of the cavity resonance. In cases where the cavity resonanceis given by a distribution that is Lorentzian, the bandwidth is given bythe full-width at half-maximum (FWHM) of the cavity frequency response.

In some implementations, the cavity of the example resonator and cavitysystem 112 has a high quality factor (a high-Q cavity), so that anelectromagnetic field in the cavity will be reflected many times beforeit dissipates. Equivalently, the photons in the cavity have a longlifetime characterized by the cavity dissipation rate κ=(ω/Q), where ωis the frequency of the wave. Such cavities can be made ofsuperconducting material and kept at cryogenic temperatures to achievequality factors that are high in value. For example, the quality factorof a high-Q cavity can have an order of magnitude in the range of10³-10⁶ or higher. Under these conditions, the electromagnetic field inthe cavity can be described quantum mechanically as being equivalent toa quantum harmonic oscillator: a standard treatment known as cavityquantum electrodynamics or cavity QED. This treatment of theelectromagnetic field in the cavity is in contrast to the Zeemaninteraction where only the spin degree of freedom is quantum mechanicalwhile the magnetic field is still classical.

For purposes of illustration, here we provide a quantum mechanicaldescription of the cavity modes. Electromagnetic waves satisfy Maxwell'sequations and both the electric field E and the magnetic field B can bedescribed in terms of a vector potential A as

${B = {\nabla{\times A}}},{E = {\frac{\partial A}{\partial t}.}}$

The vector potential itself satisfies the wave equation

${{\nabla^{2}A} = {\frac{1}{c^{2}}\frac{\partial^{2}A}{\partial t^{2}}}},$

where c is the speed of light. The wave equation has a formal solutionin the form of the Fourier series of plane waves:

A=Σ _(k)(A _(k)(t)e ^(ik·r) +A _(k)(t)e ^(−ik·r)),

where each Fourier component A_(k) (t) also satisfies the wave equation.These plane waves are ones that the cavity supports in the case ofcavity QED and by assuming A_(k)(t) has time-dependence of the formA_(k)(t)=A_(k)e^(iω) ^(k) ^(t), the electric and magnetic fields aregiven by

E _(k) =iω _(k)(A _(k) e ^(−iω) ^(k) ^(t+ik·r) −A* _(k) e ^(−iω) ^(k)^(t−ik·r)),

B _(k) =ik×(A _(k) e ^(−iω) ^(k) ^(t+ik·r) −A* _(k) e ^(−iω) ^(k)^(t−ik·r)),

where the temporal and spatial frequencies (ω_(k) and k, respectively)are related by ω_(k)=ck.

Accordingly, the energy of a single mode k is given by

${W_{k} = {{\frac{1}{2}{\int{{V\left( {{\varepsilon_{0}E_{k}^{2}} + {\mu_{0}^{- 1}B_{k}^{2}}} \right)}}}} = {2\; \varepsilon_{0}V\; \omega_{k}^{2}{A_{k} \cdot A_{k}^{*}}}}},$

where ε₀ and μ₀ are the permittivity and permeability of free spacerespectively, such that c²μ₀ε₀=1 and V is the volume of space or cavitycontaining the radiation field. By defining the vector coefficients interms of a real and imaginary part P and Q, we can express A_(k) as:

${A_{k} = {\left( {4\; \varepsilon_{0}V\; \omega_{k}^{2}} \right)^{- \frac{1}{2}}\left( {{\omega_{k}Q_{k}} + {iP}_{k}} \right)\varepsilon_{k}}},$

where ε_(k) is the polarization vector for the electromagnetic wave. Interms of Q_(k) and P_(k) the energy is given by

${W_{k} = {\frac{1}{2}\left( {P_{k}^{2} + {\omega_{k}^{2}Q_{k}^{2}}} \right)}},$

which is the form for the energy of a simple harmonic oscillator. Hence,we may treat the vectors Q_(k) and P_(k) of the electromagnetic wave asthe position and momentum vectors of the Harmonic oscillator. Thisallows us to quantize the electromagnetic field in terms of singlequanta (photons) by the standard canonical quantization of the harmonicoscillator.

We now consider the quantum treatment of a single electromagnetic modein a cavity. The Hamiltonian for the quantum harmonic oscillator may bewritten in terms of the canonical P and Q variables as

$H = {\frac{1}{2}{\left( {P^{2} + {\omega^{2}Q^{2}}} \right).}}$

We may then define operators a and a⁺, called the annihilation andcreation operators, respectively, in terms of the vectors P and Q:

${a = {\sqrt{\frac{\omega}{2\; \hslash}}\left( {Q + {\frac{i}{\omega}P}} \right)}},{a^{+} = {\sqrt{\frac{\omega}{2\; \hslash}}{\left( {Q - {\frac{i}{\omega}P}} \right).}}}$

These operators satisfy the commutation relation [a, a⁺]=1. Hence, ourHamiltonian may be written in terms of the creation and annihilationoperators as

$H = {\hslash \; {{\omega \left( {{a^{+}a} + \frac{1}{2}} \right)}.}}$

The constant factor of a half corresponds to a constant energy shift ofthe cavity modes so we may remove it by going into an interaction framewhich rescales the energies by this constant amount.

The energy eigenstates of this Hamiltonian are the so-called numberstates, which correspond to a single quanta (photon) of radiation withinthe cavity. They are labeled |n

_(c) where n=[0, 1, 2, 3, . . . ]. The action of the creation andannihilation operators on the number states is to create or remove aphoton from the cavity:

a|n

_(c) =√{square root over (n)}|n−1

_(c)

a ⁺ |n

_(c)=√{square root over (n+1)}|n+1

_(c)

Hence the operator N=a⁺a (the number operator) gives the total number ofphotons for a given number state:

a ⁺ a|n

_(c) =n|n

_(c).

The photon number state |n

_(c) is an energy eigenstate of the Hamiltonian

${{H{n\rangle}} = {\hslash \; {\omega \left( {n + \frac{1}{2}} \right)}\; {n\rangle}_{c}}},$

with energy

$\left( {n + \frac{1}{2}} \right)\hslash \; {\omega.}$

We now describe how the cavity of the example resonator and cavitysystem 112 couples to the spin ensemble containing the spins 108. Thedominant interaction is once again the spin magnetic dipole coupling tothe cavity electromagnetic fields. Therefore, we have

H ₁ =−μ·B,

and now the electromagnetic field of the cavity is treated quantummechanically. In terms of the harmonic oscillator operators the magneticfield in the cavity can be written as

${B\left( {r,t} \right)} = {\sqrt{\frac{\mu_{0}\hslash \; \omega}{2\; V}}\left( {a - a^{\dagger}} \right){u\left( {r,t} \right)}\mspace{11mu} \varepsilon}$

where ε is the propagation direction, μ₀ is the free space permeabilityconstant,  is the Planck constant, and the function u(r, t) representsthe spatial and temporal wave behavior. For some examples, we takeε={circumflex over (x)}, and the function u(r, t) takes the form

u(r,t)=u(r)cos ωt=u(y,z)cos kx cos ωt,

where u(y, z) represents the cavity magnetic field spatial profile. Inthis form, the mode volume can be expressed

$V = {\frac{\int{{{u(r)}}^{2}{^{3}r}}}{\max \;\left\lbrack {{u(r)}}^{2} \right\rbrack}.}$

As such, the mode volume is related to the spatial profile of the cavitymagnetic field, and higher spatial homogeneity in the cavity magneticfield generally produces a lower mode volume. The spin-cavityinteraction Hamiltonian then becomes

${H_{I} = {\frac{1}{2}g\; \hslash \; \left( {a - a^{\dagger}} \right)\sigma_{x}}},$

where the constant g represents the coupling strength between each spinand the cavity, and σ_(x) is the x-component spin angular momentumoperator. The coupling strength can, in some instances, be defined bythe expression

${{{{g\; \hslash} = {{{- \mu} \cdot {{{\langle 0}B}}}1}}\rangle}} = {\sqrt{\frac{\mu_{0}\gamma^{2}\hslash \; \omega}{2\; V}}{{{u(r)}}.}}$

In the example equations above, the spin-cavity coupling strength isinversely proportional to the square root of the mode volume.

The example resonator and cavity system 112 includes a resonator thatcan generate a Rabi field that is applied to the spin ensemble while thesample resides in the B₀ field 104. For example, the Rabi field can be acontinuous field or a pulsed spin-locking field. In combination with theinternal Hamiltonian of the spin system, the Rabi field can provideuniversal control of the spin ensemble. In some implementations, anymagnetic resonance experiment or pulse sequence can be implemented inthis manner. The resonator can generate the Rabi field, for example,based on signals from the control system 118, and the parameters of thefield (e.g., the phase, power, frequency, duration, etc.) can bedetermined at least partially by the signal from the control system 118.

In the plot 200 shown in FIG. 2, the vertical axis 202 represents thefrequency response of the resonator and the cavity, the horizontal axis204 represents a range of frequencies, and the curve 206 shows theresponse shape for an example implementation of the resonator and cavitysystem 112. In the example shown, the lower frequency resonance (labeledω_(s)) is that of the resonator and the higher frequency resonance(labeled ω_(c)) is that of the cavity. The quality factor (Q) of thecavity is higher than the quality factor (Q) of the resonator, and theresonance frequencies differ by the Rabi frequency (labeled Ω_(R)).

The example control system 118 can control the resonator and cavitysystem 112 of the magnetic resonance system 100 shown in FIG. 1A. Insome cases, the control system 118 can also control the cooling system120 or other components of the magnetic resonance system 100. Thecontrol system 118 is electrically coupled to, and adapted tocommunicate with, the resonator and cavity system 112. For example, thecontrol system 118 can be adapted to provide a voltage or current signalthat drives the resonator, the cavity, or both; the control system 118can also acquire a voltage or current signal from the resonator, thecavity, or both.

FIG. 1B is a schematic diagram of an example control system 150. Theexample control system 150 shown in FIG. 1B includes a controller 152, awaveform generator 154, and amplifier 156, a transmitter/receiver switch158, a receiver 160, and a signal processor 162. A control system caninclude additional or different features, and the features of a controlsystem can be configured to operate as shown in FIG. 1B or in anothermanner.

In the example shown in FIG. 1B, the example control system 150 isadapted to communicate with an external system 190. For example, theexternal system 190 can be a resonator, a cavity, or another componentof a magnetic resonance system. The control system 150 can operate basedon inputs provided by one or more external sources, including theexternal system 190 or another external source. For example, the controlsystem can receive input from an external computer, a human operator, oranother source.

The example control system 150 shown in FIG. 1B can operate in multiplemodes of operation. In a first example mode of operation, the controller152 provides a desired control operation 170 to the waveform generator154. Based on the desired control operation 170, the waveform generator154 generates a waveform 172. In some cases, the waveform generator 154also receives system model data 171, and uses the system model data 171to generate the waveform 172. The waveform 172 is received by theamplifier 156. Based on the waveform 172, the amplifier 156 generates atransmit signal 174. In this mode of operation, the transmitter/receiverswitch 158 is configured to output the transmit signal 174 to theexternal system 190.

In a second example mode of operation, the transmitter/receiver switch158 is configured to acquire a signal from the external system 190. Thecontrol system 150 can amplify, process, analyze, store, or display theacquired signal. As shown in FIG. 1B, based on the signal acquired fromthe external system 190, the transmitter/receiver switch 158 provides areceived signal 176 to the receiver 160. The receiver 160 conditions thereceived signal 176 and provides the conditioned signal 178 to thesignal processor 162. The signal processor 162 processes the conditionedsignal 178 and generates data 180. The data 180 is provided to thecontroller 152 for analysis, display, storage, or another action.

The controller 152 can be (or include) a computer or a computer system,a digital electronic controller, a microprocessor or another type ofdata-processing apparatus. The controller 152 can include memory,processors, and may operate as a general-purpose computer, or thecontroller 152 can operate as an application-specific device.

We now show an example process by which the spin ensemble in the sample110 can couple to the cavity and cool under a coherent Rabi drive. Westart with an inductively driven ensemble of non-interacting spin-½particles (represented in FIG. 1A by the spins 108) quantized in a largestatic magnetic field (represented in FIG. 1A by the B₀ field 104) andmagnetically coupled to a high-Q cavity of the resonator and cavitysystem 112. In the presence of the drive provided by the resonator ofthe resonator and cavity system 112, the spins interact with the cavityvia coherent radiative processes and the spin-cavity system can betreated quantum mechanically as a single collected magnetic dipolecoupled to the cavity. In analogy to quantum optics, we describe thespin-cavity dynamics as being generated by the Tavis-Cummings (TC)Hamiltonian. Assuming the control field to be on resonance with theLarmor frequency of the spins, the spin-cavity Hamiltonian under therotating-wave approximation (RWA) is given by H=H₀+H_(R) (t)+H_(I), with

H ₀=ω_(c) a ^(†) a+ω _(s) J _(z),

H _(R)(t)=Ω_(R) cos(ω_(s) t)J _(x), and

H _(I) =g(a ^(†) J ⁻ +aJ ₊).

As before, at (a) are the creation (annihilation) operators describingthe cavity, Ω_(R) is the strength of the drive field (Rabi frequency),ω_(c) is the resonant frequency of the cavity, ω_(s) is the Larmorresonance frequency of the spins, and g is the coupling strength of thecavity to a single spin in the ensemble in units of =1. Here we haveused the notation that

J _(α)≡Σ_(j=1) ^(N) ^(s) σ_(α) ^((j))/2

are the total angular momentum spin operators for an ensemble of N_(s)spins. The state-space V of a spin ensemble of N_(s) identical spins maybe written as the direct sum of coupled angular momentum subspaces

$V = {\oplus {\begin{matrix}\frac{N_{S}}{2} \\{J = j_{0}}\end{matrix}V_{J}^{\oplus \; n_{J}}}}$

where j₀=0 (½) if N_(s) is even (odd). V_(J) is the state space of aspin-J particle with dimension d_(J)=2J+1, and there are n_(J)degenerate subspaces with the same total spin J. Since the TCHamiltonian has a global SU(2) symmetry, it will not couple betweensubspaces in this representation. The largest subspace in thisrepresentation is called the Dicke subspace and consists of all totallysymmetric states of the spin ensemble. The Dicke subspace corresponds toa system with total angular momentum J=N_(s)/2. The TC Hamiltonianrestricted to the Dicke subspace is known as the Dicke model and hasbeen studied for quantum optics.

The eigenstates of H₀ are the tensor products of photon-number statesfor the cavity and spin states of collective angular momentum of eachtotal-spin subspace in the J_(z) direction: |n

_(c)|J, m_(z)

_(s). Here, n=0, 1, 2, . . . , m_(z)=−J, −J+1, . . . , J−1, J, and Jindexes the coupled angular momentum subspace V_(J). The collectiveexcitation number of the joint system is given byN_(ex)=a^(†)a+(J_(z)+J). The interaction term H_(I) commutes withN_(ex), and hence preserves the total excitation number of the system.This interaction can drive transitions between the state |n

_(c)|J, m_(z)

_(s) and states |n+1

_(c)|J, m_(z)−1

_(s) and |n−1

_(c)|J, m_(z)+1

_(s) at a rate of √{square root over((n+1)(J(J+1)−m_(z)(m_(z)−1)))}{square root over((n+1)(J(J+1)−m_(z)(m_(z)−1)))}{square root over((n+1)(J(J+1)−m_(z)(m_(z)−1)))} and √{square root over(n(J(J+1)−m_(z)(m_(z)+1)))}{square root over(n(J(J+1)−m_(z)(m_(z)+1)))}, respectively.

After moving into a rotating frame defined by H₁=ω_(s)(a^(†)a+J_(z)),the spin-cavity Hamiltonian is transformed to

{tilde over (H)} ⁽¹⁾ =e ^(itH) ¹ H _(sc) e ^(−itH) ¹ −H ₁,

{tilde over (H)} ⁽¹⁾ =δωa ^(†) a+Ω _(R) J _(x) +g(a ^(†) J ⁻ +aJ ₊).

Here, δω=ω_(c)−ω_(s) is the detuning of the drive from thecavity-resonance frequency, and we have made the standard rotating waveapproximation (RWA) to remove any time dependent terms in theHamiltonian.

If we now move into the interaction frame of H₂=δωa^(†)a+Ω_(R)J_(x)/2,the Hamiltonian transforms to

${{\overset{\sim}{H}}^{(2)}(t)} = {{H_{0\; \Omega_{R}}(t)} + {H_{- \Omega_{R}}(t)} + {H_{+ \Omega_{R}}(t)}}$H_(0Ω_(R))(t) = g(^(−i δ ω t)a + ^(i δ ω t)a^(†))J_(x)${H_{- \Omega_{R}}(t)} = {\frac{ig}{2}\left( {{^{{- {i(\; {{\delta \; \omega} - \Omega_{R}})}}t}{aJ}_{+}^{(x)}} - {^{{i(\; {{\delta \; \omega} - \Omega_{R}})}t}a^{\dagger}J_{-}^{(x)}}} \right)}$${H_{+ \Omega_{R}}(t)} = {\frac{ig}{2}\left( {{^{{- {i(\; {{\delta \; \omega} - \Omega_{R}})}}t}{aJ}_{-}^{(x)}} - {^{{i(\; {{\delta \; \omega} - \Omega_{R}})}t}a^{\dagger}J_{+}^{(x)}}} \right)}$

where J_(±) ^((x))≡J_(y)±iJ_(z) are the spin-ladder operators in thex-basis.

In analogy to Hartmann-Hahn matching in magnetic resonancecross-relaxation experiments for δω>0, we may set the cavity detuning tobe close to the Rabi frequency of the drive, so that Δ=δω−Ω_(R) is smallcompared to δω. By making a second rotating-wave approximation in theinteraction Hamiltonian reduces to the H_(−Ω) _(R) flip-flop exchangeinteraction between the cavity and spins in the x-basis:

${H_{I}(t)} = {\frac{ig}{2}\left( {{^{i\; \Delta \; t}{aJ}_{+}^{(x)}} - {^{i\; \Delta \; t}a^{\dagger}J_{-}^{(x)}}} \right)}$

This rotating-wave approximation is valid in the regime where thedetuning and Rabi drive strength are large compared to the time scale,t_(c), of interest (δω, Ω_(R)>>1/t_(c)). From here we will drop the (x)superscript and just note that we are working in the J_(x) eigenbasisfor our spin ensemble.

In some implementations, isolating the spin-cavity exchange interactionallows efficient energy transfer between the two systems, permittingthem to relax to a joint equilibrium state in the interaction frame ofthe control field. The coherent enhancement of the ensemble spin-cavitycoupling can enhance spin polarization in the angular momenta subspacesV_(J) at a rate greatly exceeding the thermal relaxation rate. FIG. 3shows this coherent enhancement in terms of the coupled energy levels ofthe spin-cavity states.

FIG. 3 shows two example energy level diagrams 302, 304 for a spincoupled to a two-level cavity. In both diagrams, the ket |+0

represents the ground state of the spin-cavity system (where the spinand the cavity are in their respective ground states); the ket |−1

represents the excited state of the spin-cavity system (where the spinand the cavity are in their respective excited states), and the kets |+1

and |−0

represent intermediate states. In FIG. 3, the straight arrows representcoherent oscillations, and the curved arrows represent cavitydissipation.

FIG. 3 shows that when the cavity detuning is matched to the Rabi drivestrength, energy exchange transitions between spin and cavity areenhanced. The energy level diagram 302 on the left shows the transitionswithout the coherent enhancement provided by the control drive. Theenergy level diagram 304 on the right shows the transition with thecoherent enhancement provided by the control drive when Δ=67 ω−Ω_(R) issmall compared to δω. As shown in the energy level diagram 302 on theleft, without the control drive all transition pathways are possible.The energy level diagram 304 on the right shows that when the Rabi driveis turned on and the cavity detuning is matched to the Rabi frequency,the energy exchange transitions between the spin and cavity areenhanced.

In the description below, to model the cavity-induced cooling of thespin system, we use an open quantum system description of the cavity andspin ensemble. The joint spin-cavity dynamics may be modeled using thetime-convolutionless (TCL) master equation formalism, allowing thederivation of an effective dissipator acting on the spin ensemble alone.Since the spin-subspaces V_(J) are not coupled by the TC Hamiltonian,the following derivation is provided for all values of J in thestate-space factorization.

The evolution of an example spin-cavity system can be described by theLindblad master equation

${{\frac{}{t}{\rho (t)}} = {{{\mathcal{L}_{I}(t)}{\rho (t)}} + {_{c}{\rho (t)}}}},$

where

₁ is the super-operator

₁(t)ρ=−i[H₁ (t),ρ] describing evolution under the interactionHamiltonian and

c is a dissipator describing the quality factor of the cavityphenomenologically as a photon amplitude damping channel:

$_{c} = {\frac{\kappa}{2}{\left( {{\left( {1 + \overset{\_}{n}} \right){D\lbrack a\rbrack}} + {\overset{\_}{n}{D\left\lbrack a^{\dagger} \right\rbrack}}} \right).}}$

Here, the function D[A](ρ)=2AρA^(\)−{A^(\)A,ρ}, n=tr[a^(\)aρ_(eq)]characterizes the temperature of the environment (e.g., the coolingsystem or other environment), and κ is the cavity dissipation rate(∝1/Q). The expectation value of the number operator at equilibrium isrelated to the temperature T_(c) of the environment by

${\overset{\_}{n} = {\left. \left( {^{{\omega_{c}/k_{B}}T} - 1} \right)^{- 1}\Leftrightarrow T \right. = {\frac{\omega_{c}}{k_{B}}\left\lbrack {\ln \left( \frac{1 + \overset{\_}{n}}{\overset{\_}{n}} \right)} \right\rbrack}^{- 1}}},$

where k_(B) is the Boltzmann constant.

The reduced dynamics of the spin ensemble in the interaction frame ofthe dissipator is given to second order by the TCL master equation:

${{\frac{}{t}{\rho_{s}(t)}} = {\int_{0}^{t - t_{0}}{{\tau}\; {{tr}_{c}\left\lbrack {{\mathcal{L}_{I}(t)}^{\tau \; _{c}}{\mathcal{L}_{I}\left( {t - \tau} \right)}{{\rho_{s}(t)} \otimes \rho_{eq}}} \right\rbrack}}}},$

where ρ_(s)(t)=tr_(c)[ρ(t)] is the reduced state of the spin ensembleand ρ_(eq) is the equilibrium state of the cavity. Under the conditionthat κ>>g√{square root over (N_(s))}, the master equation reduces to

${{\frac{}{t}{\rho_{s}(t)}} = {\frac{g^{2}}{4}{\int_{0}^{t - t_{0}}{{\tau}\; {^{{- \kappa}\; {\tau/2}}\left( {{{\cos \left( {\Delta \; \tau} \right)}_{s}{\rho_{s}(t)}} - {{\sin \left( {\Delta \; \tau} \right)}{\mathcal{L}_{s}(t)}{\rho (t)}}} \right)}}}}},{where}$${_{s} = {{\left( {1 + \overset{\_}{n}} \right){D\left\lbrack J_{-} \right\rbrack}} + {\overset{\_}{n}{D\left\lbrack J_{+} \right\rbrack}}}},{{\mathcal{L}_{s}\rho} = {- {\left\lbrack {H_{s},\rho} \right\rbrack}}},{and}$$H_{s} = {{\left( {1 + \overset{\_}{n}} \right)J_{+}J_{-}} - {\overset{\_}{n}\; J_{-}J_{+}}}$

are the effective dissipator and Hamiltonian acting on the spin ensembledue to coupling with the cavity.

Under the condition that κ>>g√{square root over (N_(s))} we may take theupper limit of the integral in the equation above to infinity to obtainthe Markovian master equation for the driven spin ensemble:

${\frac{}{t}{\rho_{s}(t)}} = {\left( {{\Omega_{s}\mathcal{L}_{s}} + {\frac{\Gamma_{s}}{2}_{s}}} \right){\rho_{s}(t)}}$where${\Omega_{s} = {- \frac{g^{2}\Delta}{\kappa^{2} + {4\Delta^{2}}}}},{\Gamma_{s} = {\frac{g^{2}\kappa}{\kappa^{2} + {4\Delta^{2}}}.}}$

Here, Ω_(s) is the frequency of the effective Hamiltonian, and Γ_(s) isthe effective dissipation rate of the spin-system.

We can consider the evolution of a spin state that is diagonal in thecoupled angular momentum basis, ρ(t)=Σ_(J)Σ_(m=−J) ^(J)P_(m)(t)ρ_(J,m).Here, the sum over J is summing over subspaces V_(J), and P_(J,m) (t)=

J,m|ρ(t)|J,m

is the probability of finding the system in the state ρ_(J,m)=|J,m

J,m| at time t. In this case the Markovian master equation reduces to arate equation for the populations:

${{\frac{}{t}{P_{J,m}(t)}} = {\Gamma_{s}\left( {{A_{J,{m + 1}}{P_{J,{m + 1}}(t)}} + {B_{J,m}{P_{J,m}(t)}} + {C_{J,{m - 1}}{P_{J,{m - 1}}(t)}}} \right)}},{where}$$A_{J,m} = {\left( {1 + \overset{\_}{n}} \right)\left\lbrack {{J\left( {J + 1} \right)} - {m\left( {m - 1} \right)}} \right\rbrack}$${C_{J,m} = {\overset{\_}{n}\left\lbrack {{J\left( {J + 1} \right)} - {m\left( {m + 1} \right)}} \right\rbrack}},{and}$B_(J, m) = −(A_(m) + C_(m)).

Defining {right arrow over (P)}_(J)(t)=(P_(J,−J)(t), . . . ,P_(J,J)(t)), we obtain the following matrix differential equation foreach subspace V_(J):

${{\frac{}{t}{{\overset{\rightarrow}{P}}_{J}(t)}} = {\Gamma_{s}M_{J}{{\overset{\rightarrow}{P}}_{J}(t)}}},$

where M_(J) is the tridiagonal matrix

$M_{J} = {\begin{pmatrix}B_{J,{- J}} & A_{J,{{- J} + 1}} & 0 & 0 & 0 & \ldots & 0 \\C_{J,{- J}} & B_{J,{{- J} + 1}} & A_{J,{{- J} + 2}} & 0 & 0 & \ldots & 0 \\0 & C_{J,{{- J} + 1}} & B_{J,{{- J} + 2}} & A_{J,{{- J} + 3}} & 0 & \ldots & 0 \\\vdots & \; & \; & \ddots & \; & \; & \vdots \\0 & \; & \ldots & \; & 0 & C_{J,{J - 1}} & B_{J,J}\end{pmatrix}.}$

For a given state specified by initial populations {right arrow over(P)}_(J)(0), the above differential equation has the solution {rightarrow over (P)}_(J)(t)=exp(tΓ_(s)M_(J)) P_(J)(0). The equilibrium stateof each subspace V_(J) of the driven spin ensemble satisfiesM_(J)·{right arrow over (P)}_(J)(∞)=0, and is given by

${\rho_{J,{eq}} = {\sum_{m = {- J}}^{J}{{P_{J,m}(\infty)}\rho_{J,m}}}},{where}$${P_{J,m}(\infty)} = {\frac{{{\overset{\_}{n}}^{J + m}\left( {1 + \overset{\_}{n}} \right)}^{J - m}}{\left( {1 + \overset{\_}{n}} \right)^{{2J} + 1} - {\overset{\_}{n}}^{{2J} + 1}}.}$

The total spin expectation value for the equilibrium state of the spinensemble is

${\langle J_{x}\rangle}_{eq} = {{- J} + \overset{\_}{n} - {\frac{\left( {{2J} + 1} \right){\overset{\_}{n}}^{{2J} + 1}}{\left( {1 + \overset{\_}{n}} \right)^{{2J} + 1} - {\overset{\_}{n}}^{{2J} + 1}}.}}$

If we consider the totally symmetric Dicke subspace in the limit ofN_(s)>> n, we have that the ground state population at equilibrium isgiven by P_(N) _(s) _(/2,−N) _(s) _(/2)≈1/(1+ n) and the finalexpectation value is approximately

J_(x)

_(eq)≈−N_(s)/2+ n. Thus, the final spin polarization in the Dickesubspace will be roughly equivalent to the thermal cavity polarization.

We note that, if the detuning δω were negative in the example describedabove, matching Ω_(R)=δω would result in the H_(+Ω) _(R) term beingdominant, leading to a master equation with the operators J⁻ and J₊interchanged, the dynamics of which would drive the spin ensembletowards the

J_(x)

=J state. The detuning can be made larger than the cavity linewidth toprevent competition between the H_(−Ω) _(R) and H_(+Ω) _(R) terms, whichwould drive the spin system to a high entropy thermally mixed state.

In some implementations, the cavity-resonance frequency (ω_(c)) is setbelow the spin-resonance frequency (ω_(s)) such that the detuningδω=ω_(c)−ω_(s) is a negative value. In such cases, the techniquesdescribed here can be used to perform cavity-based heating of the spinsto increase the polarization of spin ensemble. In such cases, the energyof the spin ensemble is increased by the interaction between the cavityand the spin ensemble.

The tridiagonal nature of the rate matrix allows {right arrow over(P)}_(J) _(J) (t)=exp(tΓ_(s)M_(J)){right arrow over (P)}_(J) (0) to beefficiently simulated for large numbers of spins. For simplicity we willconsider the cooling of the Dicke subspace in the ideal case where thecavity is cooled to its ground state ( n=0), and the spin ensemble istaken to be maximally mixed (i.e., P_(m)(0)=1/(2J+1) for m=−J, . . . ,J).

FIG. 4 is a plot 400 showing simulated evolution of the normalizedexpectation value of −

J_(x)(t)

/J for the Dicke subspace of an example cavity-cooled spin ensemble. Inthe plot 400, the vertical axis 402 represents a range of values of thenormalized expectation value of

J_(x)(t)

/J for the Dicke subspace, and the horizontal axis 404 represents arange of time values. In FIG. 4, the expectation values represented bythe vertical axis 402 are normalized by −J to obtain a maximum value of1, and the time variable represented by the horizontal axis 404 isscaled by the effective dissipation rate Γ_(s) for the spin ensemble.

The plot 400 includes four curves; each curve represents the simulatedexpectation value of

J, (t)

for the Dicke subspace of a spin ensemble with a different number oftotal spins N_(s), ranging from N_(s)=10² to N_(s)=10⁵. The curve 406 arepresents a spin ensemble of 10² spins; the curve 406 b represents aspin ensemble of 10³ spins; the curve 406 c represents a spin ensembleof 10⁴ spins; and the curve 406 d represents a spin ensemble of 10⁵spins.

At a value of −

J_(x)(t)

/J=1, the total angular momentum subspace of the spin ensemble iscompletely polarized to the J_(x) ground eigenstates |J, −J

. As shown in FIG. 4, the polarization of each spin ensemble increasesover time, and the polarization increases faster for the larger spinensembles. For the examples shown, the three larger spin ensembles aresubstantially fully polarized within the timescale shown in the plot400.

In some cases, the expectation value

J_(x)(t)

versus time can be fitted to an exponential to derive an effectivecooling time-constant, T_(1,eff), analogous to the thermal spin-latticerelaxation time T₁. A fit to a model given by

${- \frac{\langle{J_{x}(t)}\rangle}{J}} = {1 - {\exp \left( {- \frac{t}{T_{1,{eff}}}} \right)}}$

yields the parameters T_(1,eff)=λ(2J)^(γ)/Γ_(s) with λ=2.0406 andγ=−0.9981. This model includes an exponential rate (1/T_(1,eff)),analogous to the thermal spin-lattice relaxation process, which includesan exponential rate (1/T₁). This model can be used for an angularmomentum subspace (e.g., the Dicke subspace) or the full Hilbert space.In some instances, the effective rate (1/T_(1,eff)) is significantlyfaster than the thermal rate (1/T₁). An approximate expression for thecooling-time constant for the spin subspace V_(J) as a function of J is

${{T_{1,{eff}}(J)} \approx \frac{2}{\Gamma_{s}J_{s}}} = {\frac{2\left( {\kappa^{2} + {4\Delta^{2}}} \right)}{g^{2}\kappa \; J}.}$

In this effective cooling time-constant, the cooling efficiency ismaximized when the Rabi drive strength is matched to the cavity detuning(i.e., A=0). In this case, the cooling rate and time-constant simplifyto Γ_(s)=g²/κ and T_(1,eff)=κ/g²J, respectively. In the case where thecavity is thermally occupied, the final spin polarization is roughlyequal to the thermal cavity polarization, and for cavity temperaturescorresponding to n<√{square root over (2J)} the effective coolingconstant T_(1,eff) is approximately equal to the zero temperature value.

A magnetic resonance system can be controlled in a manner that polarizesa sample at a rate corresponding to the effective cooling constantT_(1,eff) shown above. The magnetic resonance system can be configuredaccording to the parameters that adhere to the two rotating waveapproximations used to isolate the spin-cavity exchange term H_(I)(t).For implementations where δω≈Ω_(R), the magnetic resonance system can beconfigured such that g√{square root over (N_(s))}<<κ<<Ω_(R), δω<<ω_(c),ω_(s).

For an example implementation using X-band pulsed electron spinresonance (ESR) (ω_(c)/2π≈ω_(s)/2π=10 GHz) with samples that containfrom roughly N_(s)=10⁶ spins to N_(s)=10¹⁷ spins, the magnetic resonancesystem can be configured such that Ω_(R)/2π=100 MHz, Q=10⁴ (κ/2π=1 MHz)and g/2π=1 Hz. For these parameters, the range of validity of theMarkovian master equation is N_(s)<<κ²/g²=10¹² and the Dicke subspace ofan ensemble containing roughly 10¹¹ electron spins may be polarized withan effective T₁ of 3.18 μs. This polarization time is significantlyshorter than the thermal T₁ for low-temperature spin ensembles, whichcan range from seconds to hours.

FIG. 5 is an energy level diagram 500 of an example spin system coupledto a two-level cavity. Coherent transitions are denoted by a solid lineand cavity dissipation rates are denoted by a curved line. States ineach subspace V_(J) are labeled |n

_(c)|−J_(x)+m

_(s), where m is the number of spin excitations and n is the number ofcavity excitations. Within each subspace V_(J), for cooling dynamics toappear Markovian, states of high cavity excitation number should not besignificantly populated on a coarse-grained timescale.

In the examples shown here, the spin ensemble is cooled by a coherentinteraction with the cavity, which increases the polarization of thespin ensemble. These cavity-based cooling techniques are different fromthermal T₁ relaxation, for example, because the cavity-based techniquesinclude coherent processes over the entire spin ensemble. Thermal T₁relaxation is an incoherent process that involves exchanging energybetween individual spins and the environment, which is weakly coupledwhen T₁ is long. Cavity-based cooling techniques can provide acontrolled enhancement of the spins' coupling to the thermalenvironment, by using the cavity as a link between the spin ensemble andthe environment. The cavity is more strongly coupled to the environmentthan the spin ensemble, so energy in the form of photons is dissipatedmore quickly. Due to the inherently small coupling of an individual spinto the cavity, the cavity can be efficiently coupled to the spinensemble by driving the spin ensemble so that it interacts collectivelywith the cavity as a single dipole moment with a greatly enhancedcoupling to the cavity. In some cases, the resulting link between thespin ensemble and environment—going through the cavity—is significantlystronger than the link between the spin ensemble and the environment inthe absence of the cavity, resulting in higher efficiency of energydissipation from the spin ensemble when using the cooling algorithm, anda shorter effective T₁.

The discussion above shows how the Dicke subspace and the othersubspaces are polarized by cavity-based cooling techniques. We nowdescribe how the entire state can be cooled. Due to a global SU(2)symmetry, the state space of the spin ensemble factorizes into coupledangular momentum subspaces for the spins. The largest dimension subspaceis called the Dicke subspace (which corresponds to an angular momentumJ=N/2, where N is the number of spins). For example:

${2\mspace{14mu} {spins}\text{:}\mspace{14mu} \left( {{Spin} - \frac{1}{2}} \right)^{\otimes 2}}->{{Spin} - {{1({triplet})} \oplus {Spin}} - {0({singlet})}}$${3\mspace{14mu} {spins}\text{:}\mspace{14mu} \left( {{Spin} - \frac{1}{2}} \right)^{\otimes 3}}->{{Spin} - {\frac{3}{2} \oplus {Spin}} - {\frac{1}{2} \oplus {Spin}} - {\frac{1}{2}.}}$

As shown in FIG. 6, in the 3-spins case, the spin-3/2 subspace has thelargest dimension and thus is the Dicke subspace.

FIG. 6 is a diagram 600 of an example state space represented as a3-spin Hilbert space. The diagram 600 is a matrix representation of the3-spin Hilbert space. The matrix has a block-diagonal form, where eachblock along the diagonal represents a distinct subspace. The first blockrepresents a spin-3/2 subspace 602, and the second and third blocksrepresent two spin-½ subspaces 604 a, 604 b. In FIG. 6, the spin-3/2subspace 602 is the Dicke subspace because it is the subspace of largestdimension. Cavity-based cooling can cool each respective subspace to itsrespective ground state. An interaction that breaks the SU(2) symmetryof the Hilbert space can couple the distinct subspaces, and cavity-basedcooling can cool the spin system to the true ground state of the entireHilbert space. In the example 3-spins case shown in FIG. 6, the trueground state resides in the spin-3/2 subspace 602.

Cavity-based cooling can act independently on each subspace, coolingeach subspace to its respective ground state with an effectiverelaxation time of

${T_{1,J} = \frac{1}{\Gamma_{s}J}},$

where J is the spin of the subspace, and Γ_(s) is the cavity-coolingrate derived from the Markovian master equation. In some examples, thetrue ground state of the spin ensemble is the state where all spins areeither aligned or anti-aligned with the B₀ field, and that state is inthe Dicke subspace. Generally, at thermal equilibrium the spin ensemblewill be in a mixed state, and there will be a distribution of statespopulated in all or substantially all subspaces.

The true ground state (or in some cases, another state) of the spinensemble can be reached by coupling between the spin-J subspaces. Thismay be achieved by an interaction that breaks the global SU(2) symmetryof the system Hamiltonian, for example, as described with respect toFIG. 1C. In some examples, the secular dipole-dipole interaction betweenspins, T2 relaxation, an external gradient field, or a similar externalor internal dephasing interaction is sufficient to break this symmetry.

In some implementations, applying the cooling algorithm in the presenceof a perturbation that breaks this symmetry allows cooling to the trueground state. In the case of the dipole-dipole interaction, simulationssuggest that the spins can be cooled to the true ground state at afactor of approximately √{square root over (N_(s))}/2 times the coolingrate of the Dicke subspace. This gives an effective relaxation time tothe true ground state of

$T_{1,{dipole}} = {\frac{1}{\Gamma_{s}\sqrt{N_{s}}}.}$

As in the other examples above, we consider a model that includes anexponential rate (1/T_(1,dipole)) that is analogous to the thermalspin-lattice relaxation rate (1/T₁).

FIG. 7 is a plot 700 showing effective cooling times calculated forexample spin ensembles. The plot 700 includes a log-scaled vertical axis702 showing a range of cooling times in units of seconds, and alog-scaled horizontal axis 704 showing a range of values for the numberof spins in the spin ensemble N_(s). Three curves are shown in the plot700. The curve 708 represents the cooling times for example spinensembles under the thermal T₁ relaxation process. The other two curvesrepresent the cooling times for the same example spin ensembles underthe non-thermal, coherent, cavity-based cooling processes describedabove. In particular, the curve 706 a represents the effective coolingtimes for a spin ensemble to reach the true ground state, and the curve706 b represents the effective cooling times for the Dicke subspace toreach its ground state.

FIG. 7 was generated based on a model of an electron spin ensemble in anX-band ESR system. In the model used for these calculations, theresonator and spin ensemble are both cooled to liquid heliumtemperatures (4.2 K). A typical thermal T₁ at this temperature is threeseconds for a sample of irradiated quartz. The thermal T₁ is independentof the number of spins in the sample, as shown by the curve 706 a inFIG. 7.

To obtain the curve 706 b in FIG. 7, showing the effective cooling timeconstant for the Dicke subspace of a sample subjected to cavity-basedcooling, we solved a Markovian master equation for a spin system havinga spin-resonance frequency of 10 GHz. The model used for thecalculations included a cavity-spin coupling of 1 Hz, a cavitydissipation rate of 1 MHz, a cavity detuning outside the bandwidth ofthe resonator, and a Rabi drive strength equal to this detuning. Toobtain the curve 706 a in FIG. 7, showing the effective cooling timeconstant for the full spin ensemble under cavity-based cooling withdipolar interaction, we based our results on small numbers of spins andextrapolated to larger numbers. Our initial findings suggest that

T _(1,eff)≈√{square root over (N _(s))}T _(1,Dicke).

As noted above, we consider a spin polarization model that evolvesaccording to an exponential rate (1/T_(1,eff)), which is analogous tothe thermal spin-lattice relaxation process, which evolves according toan exponential rate (1/T₁).

For the examples shown in FIG. 7, if the sample is initially restrictedto the Dicke subspace, cavity-based cooling gives a speed up overthermal T₁ for samples of greater than 10⁵ spins. If we consider acompletely mixed sample, by including a dipolar interaction whileperforming cavity-based cooling of the spin ensemble, we obtain a speedup over thermal T₁ for samples of greater than 10¹⁰ spins.

In the model for cavity-based cooling of a spin ensemble presentedabove, several assumptions are made for illustration purposes. In someinstances, the results and advantages described above can be achieved insystems that do not adhere to one or more of these assumptions. First,we have assumed that the spin ensemble is magnetically dilute such thatno coupling exists between spins. A spin-spin interaction that breaksthe global SU(2) symmetry of the Tavis-Cummings (TC) Hamiltonian willconnect the spin-J subspaces in the coupled angular momentumdecomposition of the state space. Such an interaction may be used as anadditional resource that should permit complete polarization of the fullensemble Hilbert space. Second, we have neglected the effects of thermalrelaxation of the spin system. In some instances, as the cooling effectof the cavity on the spin system relies on a coherent spin-cavityinformation exchange, the relaxation time of the spin system in theframe of the Rabi drive—commonly referred to as T_(1,ρ) —should besignificantly longer than the inverse cavity dissipation rate 1/κ.Third, we have assumed that the spin-cavity coupling and Rabi drive arespatially homogeneous across the spin ensemble. Inhomogeneities may becompensated for, for example, by numerically optimizing a control pulsethat implements an effective spin-locking Rabi drive of constantstrength over a range of spin-cavity coupling and control fieldamplitudes.

In some implementations, the ability of the cavity to remove energy fromthe spin system depends at least partially on the cooling power of thecooling system used to cool the cavity. In the example simulationspresented above, the cooling power of the cooling system is taken to beinfinite, corresponding to an infinite heat capacity of the cavity. Thetechniques described here can be implemented in a system where thecavity has a finite heat capacity. In FIGS. 8A and 8B, we give a modelof the flow of entropy and energy in an example cavity-based coolingprocess.

FIG. 8A is a schematic diagram 800 showing entropy flow in an examplecavity-based cooling process. In the diagram 800, the spins 802represent a spin ensemble, the cavity 804 represents a cavity that iscoupled to the spin ensemble, for example, under the conditionsdescribed above, and the fridge 806 represents a refrigerator or anothertype of cooling system that cools the cavity. Energy removed from thespin ensemble flows to the cavity at a rate of Γ_(SC), and energy isremoved from the cavity at a rate of Γ_(CF) by the (finite) coolingpower of the refrigerator.

FIG. 8B is a plot 810 showing example values of the dissipation ratesΓ_(SC) and Γ_(CF). The plot 810 includes a vertical axis 812representing a range of values for cooling power in units of microwatts(μW), and a log-scaled horizontal axis 814 showing a range of values forthe number of spins in the spin ensemble N_(s). Because the coolingpower of the fridge 806 is held constant in the simulations representedin the plot 810, the rate Γ_(CF) of entropy removal from the cavity tothe refrigerator remains constant, as shown by the curve 816 a. The rateΓ_(SC) of entropy removal from the spin ensemble to the cavity,represented by the curve 816 b, was calculated by specifying the totalenergy to be removed from the spin system to polarize it divided by thetime over which that energy is removed, calculated based on our derivedcooling times. The total energy removed from the spin system wascalculated as (N_(s)/2)ω, where ω was taken to be 2π10 GHz. In theexamples shown, the spin system is an electron spin ensemble that startsin the fully mixed state such that half the spins must be driven totheir ground state.

Energy deposited into the cavity is removed by the fridge at a rate thatis based on the cooling power of the fridge, which is typically on theorder of tens of microwatts (as shown in FIG. 8B) in some exampleapplications. The curve 816 b in FIG. 8B demonstrates that under someconditions, for ensembles larger than roughly 10¹³ electron spins, abottleneck of entropy flow may exist that will limit the minimum coolingtime for larger ensembles. However, in the example shown, an ensemble of10¹² electron spins may be cooled in roughly 3.18 microseconds (μs)given a fridge with cooling power of 50 μW. An ensemble of this size issufficient to obtain a strong electron spin resonance signal.

Finally, the derivation of the Markovian master equation above assumesthat no correlations between cavity and spin system accrue during thecooling process, such that there is no back action of the cavitydynamics on the spin system. This condition is enforced when the cavitydissipation rate, κ, exceeds the rate of coherent spin-cavity exchangein the lowest excitation manifold by at least an order of magnitude(i.e. κ>10g√{square root over (N_(s))}). In this Markovian limit, therate at which spin photons are added to the cavity is significantly lessthan the rate at which thermal photons are added, meaning the coolingpower of the fridge necessary to maintain the thermal cavity temperatureis sufficient to dissipate the spin photons without raising the averageoccupation number of the cavity. From the above equation we see that thecooling efficiency could be improved by adding more spins to make κcloser to g√{square root over (N_(s))}; in this regime the cooling powerof the fridge may not be sufficient to prevent back action from thecavity and non-Markovian effects significantly lower the cooling rate.

While this specification contains many details, these should not beconstrued as limitations on the scope of what may be claimed, but ratheras descriptions of features specific to particular examples. Certainfeatures that are described in this specification in the context ofseparate implementations can also be combined. Conversely, variousfeatures that are described in the context of a single implementationcan also be implemented in multiple embodiments separately or in anysuitable subcombination.

Example implementations of several independent, general concepts havebeen described. In one general aspect of what is described above, adrive field is applied to a spin ensemble in a static magnetic field.The drive field is adapted to couple spin states of the spin ensemblewith one or more cavity modes of a cavity. Polarization of the spinensemble is increased by the coupling between the spin states and thecavity mode.

In another general aspect of what is described above, a cavity iscoupled with a spin ensemble in a sample. The sample can be held at athermal temperature and subject to a static magnetic field, and aninteraction between the cavity and the spin ensemble is generated (e.g.,by applying a drive field). The interaction increases polarization ofthe spin ensemble faster than the internal polarizing process affectingthe sample.

In some implementations of the general concepts described above,polarization of the spin ensemble is increased by cavity-based coolingacting independently on each angular momentum subspace of the spinensemble via the coupling between the spin states and the cavity mode,and a mixing process mixing the angular momentum subspaces. Theoperations can be applied iteratively in some instances. The angularmomentum subspaces can be mixed, for example, by a dipolar interaction,a transverse (T₂) relaxation process, application of a gradient field,or a combination of these and other processes.

In some implementations of the general concepts described above, thecavity has a low mode volume and a high quality factor. The mode volume,the quality factor, or a combination of these and other cavityparameters can be designed to produce a coupling between the spinensemble and the cavity that effectively “short-circuits” the spinensemble polarization process. In some examples, the cavity has a modevolume V and a quality factor Q, such that κ>>g√{square root over(N_(s))}. Here, N_(s) represents the number of spins in the spinensemble, κ=(ω_(c)/Q) represents the dissipation rate of the cavity,ω_(c) represents the resonance frequency of the cavity, and g representsthe coupling strength of the cavity to an individual spin in the spinensemble. In some examples, the dissipation rate κ is more than twotimes g√{square root over (N_(s))}. In some examples, the dissipationrate κ is an order of magnitude greater than g√{square root over(N_(s))}. In some examples, the dissipation rate κ is two or threeorders of magnitude greater than g√{square root over (N_(s))}. In someinstances, the coupling between the spin ensemble and the cavityincreases polarization of the spin ensemble faster than the thermalspin-lattice (T₁) relaxation process.

In some implementations of the general concepts described above, thespin ensemble has a spin-resonance frequency (ω_(s)), and the drivefield is generated by a resonator that is on-resonance with thespin-resonance frequency (ω_(s)). The drive field can be a time-varying(e.g., oscillating or otherwise time-varying) magnetic field. In somecases, the spin ensemble is a nuclear spin ensemble, and the drive fieldis a radio-frequency field. In some cases, the spin ensemble is anelectron spin ensemble, and the drive field is a microwave-frequencyfield.

In some implementations of the general concepts described above, thecavity mode corresponds to a cavity-resonance frequency (ω_(c)), and thecavity-resonance frequency (ω_(c)) is detuned from the spin-resonancefrequency (ω_(s)) by an amount δω=ω_(c)−ω_(s). The drive field can havea drive field strength that generates Rabi oscillations at a Rabifrequency (Ω_(R)). In some cases, the detuning δω is substantially equalto Ω_(R). For instance, the difference Δ=δω−Ω_(R) can be small comparedto the detuning δω. In some examples, the difference Δ is less than halfthe detuning δω. In some examples, the difference Δ is an order ofmagnitude less than the detuning δω. In some examples, the difference Δis two or three orders of magnitude less than the detuning δω.

In some implementations of the general concepts described above, theinteraction between the cavity and the spin ensemble increasespolarization of the spin ensemble at a polarization rate that is relatedto a parameter of the cavity. In some instances, the polarization ratecan be higher or lower due to an electromagnetic property of the cavity,such as the value of the quality factor, the value of the mode volume,the value of the dissipation rate, or another property. In some cases,the polarization rate is related to a coupling strength g between thecavity and a spin in the spin ensemble. As an example, the polarizationrate can be related to the dissipation rate

${\Gamma_{s} = \frac{g^{2}\kappa}{\kappa^{2} + {4\Delta^{2}}}},$

where κ represents a dissipation rate of the cavity, g represents thecoupling strength of the cavity to a spin in the spin ensemble, andΔ=δω−Ω_(R). In some cases, the polarization rate is also related to thenumber of spins in the spin ensemble N_(s).

In some implementations of the general concepts described above, thestatic magnetic field is applied to the spin ensemble by a primarymagnet system, and the static magnetic field is substantially uniformover the spin ensemble. The drive field can be oriented orthogonal tothe static magnetic field. For example, the static magnetic field can beoriented along a z-axis, and the drive field can be oriented in thexy-plane (which is orthogonal to the z-axis).

In some implementations of the general concepts described above, heatenergy is removed from the cavity by operation of a cooling system thatresides in thermal contact with the cavity. The cooling system can coolthe cavity. In some cases, the spin ensemble dissipates photons to thecooling system, or to another thermal environment of the cavity, throughthe coupling between the spin states and the cavity mode.

In some implementations of the general concepts described above, thedrive field is generated by a resonator. In some cases, the resonatorand cavity are formed as a common structure or subsystem. For example,the resonator and cavity can be integrated in a common, multi-moderesonator structure. In some cases, the resonator and cavity are formedas two or more separate structures. For example, the resonator can be acoil structure having a first resonance frequency, and the cavity can bea distinct cavity structure that has a second, different resonancefrequency. The resonator, the cavity, or both can includesuperconducting material and other materials.

In some implementations of the general concepts described above, thecoupling between the spin ensemble and the cavity changes the state ofthe spin ensemble. For example, the coupling can map the spin ensemblefrom an initial (mixed) state to a subsequent state that has higherpolarization than the initial state. The subsequent state can be a mixedstate or a pure state. In some cases, the subsequent state has a puritythat is equal to the purity of the cavity. In some instances, thecoupling can evolve the spin ensemble from an initial state to thethermal equilibrium state of the spin ensemble. The thermal equilibriumstate is typically defined, at least partially, by the sampleenvironment (including the sample temperature and the static magneticfield strength). In some instances, the coupling can evolve the spinensemble from an initial state to a subsequent state having apolarization that is less than, equal to, or greater than the thermalequilibrium polarization.

In some implementations of the general concepts described above, thedrive field is adapted to couple the Dicke subspace of the spin ensemblewith the cavity modes. In some representations of the spin ensemble, theDicke subspace can be defined as the largest angular momentum subspace,such that the Dicke subspace contains all the totally-symmetric statesof the spin ensemble. In some representations, the Dicke subspacecorresponds to a system with total angular momentum J=N_(s)/2, whereN_(s) is the number of spins in the spin ensemble. In some cases, theDicke subspace and multiple other angular momentum subspaces of the spinensemble are coupled with the cavity modes. In some cases, all angularmomentum subspaces of the spin ensemble are coupled with the cavitymodes.

In some implementations of the general concepts described above, theinteraction between the cavity and the spin ensemble causes the spinensemble to dissipate photons to a thermal environment via the cavitymodes. The interaction can include a coherent radiative interactionbetween the cavity and the spin ensemble. In some cases, the coherentradiative interaction can increase the spin ensemble's polarizationfaster than any incoherent thermal process (e.g., thermal spin-latticerelaxation, spontaneous emission, etc.) affecting the spin ensemble. Insome cases, the interaction drives the spin ensemble so that itinteracts collectively with the cavity as a single dipole moment.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications can be made. Accordingly, otherembodiments are within the scope of the following claims.

1. A method of controlling a spin ensemble comprising: coupling a cavitywith a spin ensemble in a sample environment, the sample environmentcomprising a static magnetic field; and generating an interactionbetween the cavity and the spin ensemble that increases polarization ofthe spin ensemble faster than the dominant thermal polarizing processaffecting the spin ensemble in the sample environment.
 2. The method ofclaim 1, comprising increasing polarization of the spin ensemble byiteratively: acting on angular momentum subspaces of the spin ensembleby the interaction between the spin ensemble and the cavity; and mixingthe angular momentum subspaces.
 3. The method of claim 2, where theangular momentum subspaces are mixed by at least one of a dipolarinteraction, a transverse (T₂) relaxation process, or application of agradient field.
 4. The method of claim 1, comprising controlling atemperature of the cavity by operation of a cooling system that isthermally coupled to the cavity.
 5. The method of claim 1, wherein theinteraction between the cavity and the spin ensemble cools the spinensemble.
 6. The method of claim 1, wherein the interaction between thecavity and the spin ensemble heats the spin ensemble.
 7. The method ofclaim 1, wherein generating the interaction comprises applying a drivefield to the spin ensemble, the drive field being adapted to couple spinstates of the spin ensemble with a cavity mode of the cavity.
 8. Themethod of claim 7, wherein the drive field is tuned to a spin-resonancefrequency (ω_(s)) of the spin ensemble, the cavity mode corresponds to acavity-resonance frequency (ω_(c)), and the cavity-resonance frequency(ω_(c)) is detuned from the spin-resonance frequency (ω_(s)) by anamount δω=ω_(c)−ω_(s).
 9. The method of claim 1, wherein the dominantthermal polarizing process comprises thermal relaxation having acharacteristic relaxation time T₁, and the interaction between thecavity and the spin ensemble increases polarization of the spin ensembleat a polarization rate that is related to a parameter of the cavity andis faster than the characteristic relaxation time T₁.
 10. A magneticresonance system comprising: a cavity adapted to interact with a spinensemble in a static magnetic field; and a resonator adapted to generatean interaction between the cavity and the spin ensemble that increasespolarization of the spin ensemble faster than the dominant thermalpolarizing process affecting the spin ensemble.
 11. (canceled)
 12. Thesystem of claim 10, wherein the resonator and the cavity are distinctstructures.
 13. The system of claim 10, comprising an integratedmulti-mode resonator structure that includes the resonator and thecavity.
 14. The system of claim 10, further comprising: a primary magnetsystem adapted to generate the static magnetic field; and a samplecontaining the spin ensemble.
 15. The system of claim 10, furthercomprising a cooling system thermally coupled to the cavity and adaptedto cool the cavity.
 16. The system of claim 15, wherein the coolingsystem comprises at least one of a liquid nitrogen cryostat, a liquidhelium cryostat, a closed-loop refrigerator, a pumped-helium cryostat, ahelium-3 refrigerator, or a dilution refrigerator.
 17. The system ofclaim 15, wherein the cooling system is thermally coupled to a samplecontaining the spin ensemble, and the cooling system is adapted to coolthe sample.